Changing times

We were 75 minutes into class and her engagement and grasp of the content was wonderful! 😊 She had, with minimal guidance, derived the formula for the length of an arc of a circle by applying proportional reasoning. When a student makes a cross-topic connection like this, it is rewarding! 😇 Now, we were solving questions to cement her understanding of the concept.

We reached the last question in the set: The second hand of a clock is 10 cm long. How far does the tip of the second hand travel in 20 seconds?

She racked her brains for close to 10 minutes before concluding that she was stumped! 😵 I asked her to share where she was getting stuck. She replied that she didn't know the speed at which the hand was moving - so, she couldn't find the distance that the tip had travelled.

I realised that she was thinking of the {speed = distance ÷ time} formula. So, I gave her a hint that knowing the speed at which the hand was moving wasn't required in this case. In addition, I asked her to visualise a clock as that could help her arrive at a logical solution.

5 more minutes ticked by... She told me that she was done; so, I asked her to explain her thought process.

Sir, I took θ as 20° and r as 10 cm and substituted it in the formula we just learnt. I got the arc length to be about 3.5 cm which is the same as the distance travelled by the tip of the hand.

When asked how she deduced the value of θ, she said that the second hand moves 1° in 1 second; on further probing, she admitted that she wasn't confident about her reasoning but went ahead with it as it was a number in the question (classic approach taken by most children when they aren't sure what to do! 😅).

At this point, I decided to do a screenshare of an image of a clock with all 3 hands (hour, minute, second) visible and of standard relative thicknesses and lengths (see image below).

(Source: https://www.titan.co.in/dw/image/v2/BKDD_PRD/on/demandware.static/-/Sites-titan-master-catalog/default/dw552c9d4b/images/Titan/Catalog/W0075PS01_1.jpg)

I asked her to tell me the number that the second hand was pointing towards. After a moment of hesitation, she said '2' - that's when the root cause of her struggle dawned on me! I enquired into whether her family had a clock at home; she said no. She did recall seeing clocks at some public places but hadn't needed to fluently read them as she relied on phones. So, I changed tack - instead of lengths of arcs, we spent some time understanding how a clock works, what the different divisions on the circumference of the clock represent and the angles swept by the different hands in 1 second, 1 minute and 1 hour.

Now, factoring in her new knowledge, I asked her to try the same question. After 2-3 minutes, she correctly reasoned that the second hand sweeps an angle of 120° (not 20°!) in 20 seconds and solved the question.

~ o ~ x ~ o ~

Here was a 13-year-old child who has shown admirable growth over the 2+ years that I've taught her, who has the resources to attend a reputed international school and who takes part in extracurricular activities/sports - basically, a child that one would consider rounded. If this is her plight with regards to clocks, I wonder if other children are in a similar boat...?

This segment of the class was eye-opening. I had assumed that she knew the parts of a clock and how it practically works. The object, be it in the form of wall clocks at home/school or as a wristwatch around my arm, was ubiquitous while I was growing up. I had internalised the way a clock works through repeated exposure. That wasn't the case for my student; hence, reasoning about how the hands on a clock move was complex for her!

Moving forward, I need to be even more careful/aware about what prior knowledge my students possess! 🫥